The Anti-de-Sitter (AdS) space is the analogue in Lorentzian geometry to the hyperbolic space of negative constant curvature familiar to Riemannian geometers. The study of the Anti-de-Sitter space and more generally, of solutions to the Einstein equations which ressemble the Anti-de-Sitter space has many applications in high energy physics and provides mathematicians with problems lying at the intersection of Lorenzian geometry and hyperbolic pdes. I will start with a general introduction to the study of the Einstein equations, with an emphasis on the evolution problem. I will then briefly discuss the well-posedness for wave equations, including the Einstein equations, in the case of AdS type boundaries. In the second part of the talk, I will present several results concerning the linear or non-linear perturbations of the AdS space for the vacuum Einstein equations and some related toy models.